Connection (algebraic framework)

Geometry of quantum systems (e.g., noncommutative geometry and supergeometry) is mainly phrased in algebraic terms of modules and algebras. Connections on modules are generalization of a linear connection on a smooth vector bundle $E\to X$ written as a Koszul connection on the $C^\infty(X)$ -module of sections of $E\to X$ .^[1]

Commutative algebra

Let $A$ be a commutative ring and $P$ an A-module. There are different equivalent definitions of a connection on $P$ .^[2] Let $D(A)$ be the module of derivations of a ring $A$ . A connection on an A-module $P$ is defined as an A-module morphism

\nabla:D(A)\ni u\to \nabla_u\in \mathrm{Diff}_1(P,P)

such that the first order differential operators $\nabla_u$ on $P$ obey the Leibniz rule

\nabla_u(ap)=u(a)p+a\nabla_u(p), \quad a\in A, \quad p\in P.

Connections on a module over a commutative ring always exist.

The curvature of the connection $\nabla$ is defined as the zero-order differential operator

R(u,u')=[\nabla_u,\nabla_{u'}]-\nabla_{[u,u']} \,

on the module $P$ for all $u,u'\in D(A)$ .

If $E\to X$ is a vector bundle, there is one-to-one correspondence between linear connections $\Gamma$ on $E\to X$ and the connections $\nabla$ on the $C^\infty(X)$ -module of sections of $E\to X$ . Strictly speaking, $\nabla$ corresponds to the covariant differential of a connection on $E\to X$ .

Graded commutative algebra

The notion of a connection on modules over commutative rings is straightforwardly extended to modules over a graded commutative algebra.^[3] This is the case of superconnections in supergeometry of graded manifolds and supervector bundles. Superconnections always exist.

Noncommutative algebra

If $A$ is a noncommutative ring, connections on left and right A-modules are defined similarly to those on modules over commutative rings.^[4] However these connections need not exist.

In contrast with connections on left and right modules, there is a problem how to define a connection on an R-S-bimodule over noncommutative rings R and S. There are different definitions of such a connection.^[5] Let us mention one of them. A connection on an R-S-bimodule $P$ is defined as a bimodule morphism

\nabla:D(A)\ni u\to \nabla_u\in \mathrm{Diff}_1(P,P)

which obeys the Leibniz rule

\nabla_u(apb)=u(a)pb+a\nabla_u(p)b +apu(b), \quad a\in R, \quad b\in S, \quad p\in P.

Notes

↑ Koszul (1950)
↑ Koszul (1950), Mangiarotti (2000)
↑ Bartocci (1991), Mangiarotti (2000)
↑ Landi (1997)
↑ Dubois-Violette (1996), Landi (1997)

References

Koszul, J., Homologie et cohomologie des algebres de Lie,Bulletin de la Societe Mathematique 78 (1950) 65
Koszul, J., Lectures on Fibre Bundles and Differential Geometry (Tata University, Bombay, 1960)
Bartocci, C., Bruzzo, U., Hernandez Ruiperez, D., The Geometry of Supermanifolds (Kluwer Academic Publ., 1991) ISBN 0-7923-1440-9
Dubois-Violette, M., Michor, P., Connections on central bimodules in noncommutative differential geometry, J. Geom. Phys. 20 (1996) 218. arXiv:q-alg/9503020v2
Landi, G., An Introduction to Noncommutative Spaces and their Geometries, Lect. Notes Physics, New series m: Monographs, 51 (Springer, 1997) ArXiv eprint, iv+181 pages.
Mangiarotti, L., Sardanashvily, G., Connections in Classical and Quantum Field Theory (World Scientific, 2000) ISBN 981-02-2013-8

External links

Sardanashvily, G., Lectures on Differential Geometry of Modules and Rings (Lambert Academic Publishing, Saarbrücken, 2012); arXiv: 0910.1515

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